>GAMES FROM HDD ON FAT PS2
The easiest way to play games is to get a fat PS2, because it's easy to mount an internal hard drive.
>HARD DRIVE FOR FAT PS2
OPTION 1: OEM SONY NETWORK ADAPTER with IDE HDD.
The original OEM Sony network adapter has working network support but only supports IDE Hard Drives. This is a simple setup but IDE hard drives are all slow spinning rust, small capacity, are probably going to die soon, and are more expensive per GiB than modern SATA drives since they stopped making IDE drives a long time ago.
OPTION 2: CHINESE SATA NETWORK ADAPTER
These things work just fine and provide SATA compatibility out of the box. They don't come with a working RJ45 port though. There's still a port on the back but I think it's dialup/phone line or something. I've used one of these in the past and it works just fine. I've never used the network for anything so I wasn't bothered by the lack of network.
OPTION 3: OEM SONY NETWORK ADAPTER WITH INTERNAL MODDED SATA BOARD
I use this now, pretty cool. It's a little internal board that goes into the OEM adapter and replaces the IDE interface with a SATA one. I think these boards are pretty recent, I don't remember them being around as of a couple years ago. Pic related.
OPTION 4: OEM SONY NETWORK ADAPTER WITH EXTERNAL IDE->SATA board.
This was the happening thing a couple years ago, but I never managed to get it to work. I bought the right startech adapter, modded the power cable etc, and still nothing, drive not recognized. The physical constraints make this a pain in the ass and it never worked for me anyways so I can't recommend it.